A341414 -id:A341414 - cp's OEIS frontend

A341616 Table read by ascending antidiagonals: T(n,j) = Fibonacci(n)*Lucas(n+j), product of the n-th term in the Fibonacci sequence (with F(1)=1 and F(2)=1) and the (n+j)-th term in the Lucas sequence (with L(1)=1 and L(2)=3 and j=0,1,2,...).

1, 3, 3, 8, 4, 4, 21, 14, 7, 7, 55, 33, 22, 11, 11, 144, 90, 54, 36, 18, 18, 377, 232, 145, 87, 58, 29, 29, 987, 611, 376, 235, 141, 94, 47, 47, 2584, 1596, 988, 608, 380, 228, 152, 76, 76, 6765, 4182, 2583, 1599, 984, 615, 369, 246, 123, 123

Offset: 1

Jens Rasmussen, Feb 16 2021

j is the offset when combining terms from the two initial sequences.

			T(4,3) = Fibonacci(4)*Lucas(4+3) = 3*29 = 87.
Square array showing T(n,j) begins:
      j=0 j=1 j=2 j=3 j=4  ..
  n=1   1   3   4   7  11  ..
  n=2   3   4   7  11  18  ..
  n=3   8  14  22  36  58  ..
  n=4  21  33  54  87 141  ..
  ...  ..  ..  ..  ..  ..  ..

For j=0 the resulting sequence is used as input in A341414.

Cf. A000045, A000032, A001622, A094214, A104457.

T(n,j) = fibonacci(2*n+j) - (-1)^n*fibonacci(j);
matrix(7,7,n,k, T(n,k-1)) \\ Michel Marcus, Mar 02 2021

For phi=(1+sqrt(5))/2 and tau=(1-sqrt(5))/2:
T(n,j) = Fibonacci(n)*Lucas(n+j).
T(n,j) = (phi^n - tau^n)*(phi^(n+j) + tau^(n+j))/sqrt(5).
T(n,j) = Fibonacci(2n+j) - (-1)^n*Fibonacci(j).
Lim_{n, j -> oo} T(n+1,j)/T(n,j) = phi^2 (A104457).
Lim_{n, j -> oo} T(n,j+1)/T(n,j) = phi (A001622).

cp's OEIS Frontend

A341616 Table read by ascending antidiagonals: T(n,j) = Fibonacci(n)*Lucas(n+j), product of the n-th term in the Fibonacci sequence (with F(1)=1 and F(2)=1) and the (n+j)-th term in the Lucas sequence (with L(1)=1 and L(2)=3 and j=0,1,2,...).

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